On the Diameter and Girth of an Annihilating-Ideal Graph

TL;DR

This paper investigates the properties of annihilating-ideal graphs of commutative rings, focusing on their diameter, girth, bipartiteness, and the relationship with zero-divisor graphs and Smarandache vertices.

Contribution

It provides characterizations of rings based on the diameter and girth of their annihilating-ideal graphs, including conditions for bipartiteness and the connection with zero-divisor graphs.

Findings

Characterized rings with annihilating-ideal graph girth at least 4.

Identified conditions under which annihilating-ideal graphs are bipartite.

Explored the relationship between Smarandache vertices and the diameter of the graph.

Abstract

Let $R$ be a commutative ring with $1\neq 0$ and $\Bbb{A}(R)$ be the set of ideals with nonzero annihilators. The annihilating-ideal graph of $R$ is defined as the graph $\Bbb{AG}(R)$ with the vertex set $\Bbb{A}(R)^{*} = \Bbb{A}(R)\setminus \{(0)\}$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ = (0)$. In this paper, we first study the interplay between the diameter of annihilating-ideal graphs and zero-divisor graphs. Also, we characterize rings $R$ when ${\rm gr}(\Bbb{AG}(R))\geq 4$, and so we characterize rings whose annihilating-ideal graphs are bipartite. Finally, in the last section we discuss on a relation between the Smarandache vertices and diameter of $\Bbb {AG}(R)$.